Ebook: Quaternion and Clifford Fourier Transforms and Wavelets

Quaternion and Clifford Fourier and wavelet transformations generalize the classical theory to higher dimensions and are becoming increasingly important in diverse areas of mathematics, physics, computer science and engineering. This edited volume presents the state of the art in these hypercomplex transformations. The Clifford algebras unify Hamilton’s quaternions with Grassmann algebra. A Clifford algebra is a complete algebra of a vector space and all its subspaces including the measurement of volumes and dihedral angles between any pair of subspaces. Quaternion and Clifford algebras permit the systematic generalization of many known concepts.

This book provides comprehensive insights into current developments and applications including their performance and evaluation. Mathematically, it indicates where further investigation is required. For instance, attention is drawn to the matrix isomorphisms for hypercomplex algebras, which will help readers to see that software implementations are within our grasp.

It also contributes to a growing unification of ideas and notation across the expanding field of hypercomplex transforms and wavelets. The first chapter provides a historical background and an overview of the relevant literature, and shows how the contributions that follow relate to each other and to prior work. The book will be a valuable resource for graduate students as well as for scientists and engineers.

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Quaternion and Clifford Fourier and wavelet transformations generalize the classical theory to higher dimensions and are becoming increasingly important in diverse areas of mathematics, physics, computer science and engineering. This edited volume presents the state of the art in these hypercomplex transformations. The Clifford algebras unify Hamilton’s quaternions with Grassmann algebra. A Clifford algebra is a complete algebra of a vector space and all its subspaces including the measurement of volumes and dihedral angles between any pair of subspaces. Quaternion and Clifford algebras permit the systematic generalization of many known concepts.

This book provides comprehensive insights into current developments and applications including their performance and evaluation. Mathematically, it indicates where further investigation is required. For instance, attention is drawn to the matrix isomorphisms for hypercomplex algebras, which will help readers to see that software implementations are within our grasp.

It also contributes to a growing unification of ideas and notation across the expanding field of hypercomplex transforms and wavelets. The first chapter provides a historical background and an overview of the relevant literature, and shows how the contributions that follow relate to each other and to prior work. The book will be a valuable resource for graduate students as well as for scientists and engineers.

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Quaternion and Clifford Fourier and wavelet transformations generalize the classical theory to higher dimensions and are becoming increasingly important in diverse areas of mathematics, physics, computer science and engineering. This edited volume presents the state of the art in these hypercomplex transformations. The Clifford algebras unify Hamilton’s quaternions with Grassmann algebra. A Clifford algebra is a complete algebra of a vector space and all its subspaces including the measurement of volumes and dihedral angles between any pair of subspaces. Quaternion and Clifford algebras permit the systematic generalization of many known concepts.

This book provides comprehensive insights into current developments and applications including their performance and evaluation. Mathematically, it indicates where further investigation is required. For instance, attention is drawn to the matrix isomorphisms for hypercomplex algebras, which will help readers to see that software implementations are within our grasp.

It also contributes to a growing unification of ideas and notation across the expanding field of hypercomplex transforms and wavelets. The first chapter provides a historical background and an overview of the relevant literature, and shows how the contributions that follow relate to each other and to prior work. The book will be a valuable resource for graduate students as well as for scientists and engineers.

Content:
Front Matter....Pages i-xxvii
Front Matter....Pages 1-1
Quaternion Fourier Transform: Re-tooling Image and Signal Processing Analysis....Pages 3-14
The Orthogonal 2D Planes Split of Quaternions and Steerable Quaternion Fourier Transformations....Pages 15-39
Quaternionic Spectral Analysis of Non-Stationary Improper Complex Signals....Pages 41-56
Quaternionic Local Phase for Low-level Image Processing Using Atomic Functions....Pages 57-83
Bochner’s Theorems in the Framework of Quaternion Analysis....Pages 85-104
Bochner–Minlos Theorem and Quaternion Fourier Transform....Pages 105-120
Front Matter....Pages 121-121
Square Roots of –1 in Real Clifford Algebras....Pages 123-153
A General Geometric Fourier Transform....Pages 155-176
Clifford–Fourier Transform and Spinor Representation of Images....Pages 177-195
Analytic Video (2D + t) Signals Using Clifford–Fourier Transforms in Multiquaternion Grassmann–Hamilton–Clifford Algebras....Pages 197-219
Generalized Analytic Signals in Image Processing: Comparison, Theory and Applications....Pages 221-246
Colour Extension of Monogenic Wavelets with Geometric Algebra: Application to Color Image Denoising....Pages 247-268
Seeing the Invisible and Maxwell’s Equations....Pages 269-284
A Generalized Windowed Fourier Transform in Real Clifford Algebra Cl 0,n ....Pages 285-298
The Balian–Low Theorem for the Windowed Clifford–Fourier Transform....Pages 299-319
Sparse Representation of Signals in Hardy Space....Pages 321-332
Back Matter....Pages 333-338

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Quaternion and Clifford Fourier and wavelet transformations generalize the classical theory to higher dimensions and are becoming increasingly important in diverse areas of mathematics, physics, computer science and engineering. This edited volume presents the state of the art in these hypercomplex transformations. The Clifford algebras unify Hamilton’s quaternions with Grassmann algebra. A Clifford algebra is a complete algebra of a vector space and all its subspaces including the measurement of volumes and dihedral angles between any pair of subspaces. Quaternion and Clifford algebras permit the systematic generalization of many known concepts.

This book provides comprehensive insights into current developments and applications including their performance and evaluation. Mathematically, it indicates where further investigation is required. For instance, attention is drawn to the matrix isomorphisms for hypercomplex algebras, which will help readers to see that software implementations are within our grasp.

It also contributes to a growing unification of ideas and notation across the expanding field of hypercomplex transforms and wavelets. The first chapter provides a historical background and an overview of the relevant literature, and shows how the contributions that follow relate to each other and to prior work. The book will be a valuable resource for graduate students as well as for scientists and engineers.

Content:
Front Matter....Pages i-xxvii
Front Matter....Pages 1-1
Quaternion Fourier Transform: Re-tooling Image and Signal Processing Analysis....Pages 3-14
The Orthogonal 2D Planes Split of Quaternions and Steerable Quaternion Fourier Transformations....Pages 15-39
Quaternionic Spectral Analysis of Non-Stationary Improper Complex Signals....Pages 41-56
Quaternionic Local Phase for Low-level Image Processing Using Atomic Functions....Pages 57-83
Bochner’s Theorems in the Framework of Quaternion Analysis....Pages 85-104
Bochner–Minlos Theorem and Quaternion Fourier Transform....Pages 105-120
Front Matter....Pages 121-121
Square Roots of –1 in Real Clifford Algebras....Pages 123-153
A General Geometric Fourier Transform....Pages 155-176
Clifford–Fourier Transform and Spinor Representation of Images....Pages 177-195
Analytic Video (2D + t) Signals Using Clifford–Fourier Transforms in Multiquaternion Grassmann–Hamilton–Clifford Algebras....Pages 197-219
Generalized Analytic Signals in Image Processing: Comparison, Theory and Applications....Pages 221-246
Colour Extension of Monogenic Wavelets with Geometric Algebra: Application to Color Image Denoising....Pages 247-268
Seeing the Invisible and Maxwell’s Equations....Pages 269-284
A Generalized Windowed Fourier Transform in Real Clifford Algebra Cl 0,n ....Pages 285-298
The Balian–Low Theorem for the Windowed Clifford–Fourier Transform....Pages 299-319
Sparse Representation of Signals in Hardy Space....Pages 321-332
Back Matter....Pages 333-338
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